Natural number

A nat­u­ral num­ber is a num­ber like 0, 1, 2, 3, 4, 5, 6, … which can be used to rep­re­sent the count of an ob­ject. The set of nat­u­ral num­bers is \(\mathbb N.\) Not all sources in­clude 0 in \(\mathbb N.\)

Nat­u­ral num­bers are per­haps the sim­plest type of num­ber. They don’t in­clude nega­tive num­bers, frac­tional num­bers, ir­ra­tional num­bers, imag­i­nary num­bers, or any of those com­plex­ities.

Thanks to their sim­plic­ity, nat­u­ral num­bers are of­ten the first math­e­mat­i­cal con­cept taught to chil­dren. Nat­u­ral num­bers are equipped with a no­tion of ad­di­tion ($2 + 3 = 5$ and so on) and mul­ti­pli­ca­tion ($2 \cdot 3 = 6$ and so on), these are among the first math­e­mat­i­cal op­er­a­tions taught to chil­dren.

De­spite their sim­plic­ity, the nat­u­ral num­bers are a ubiquitous and use­ful math­e­mat­i­cal ob­ject. They’re quite use­ful for count­ing things. They rep­re­sent all the pos­si­ble car­di­nal­ities of finite sets. They’re also a use­ful data struc­ture, in that num­bers can be used to en­code all sorts of data. Al­most all of mod­ern math­e­mat­ics can be built out of nat­u­ral num­bers.

Add a “for­mal­iza­tion” lens with the Peano ax­ioms. I recom­mend at least one page with just the raw Peano ax­ioms (and very lit­tle prose), and an­other gen­tler in­tro­duc­tion sort of like http://​​​​29glDrR and http://​​​​29nKYAL, albeit more to-the-point and prob­a­bly with­out go­ing all the way up to non-stan­dard num­ber ter­ri­tory.


  • Graham's number

    A fairly large num­ber, as num­bers go.

  • A googolplex

    A mod­er­ately large num­ber, as large num­bers go.

  • A googol

    A pretty small large num­ber.

  • Prime number

    The prime num­bers are the “build­ing blocks” of the count­ing num­bers.


  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.